Stability and robustness of a railway timetable are essential properties for punctual and reliable operations. Timetable performance evaluation is therefore an important aspect in the timetable design process. In particular, the stability and recoverability properties of a timetable with respect to daily process time variations must be well analysed. The timetable must be able to recover from primary delays due to stochastic process times and it must be robust against secondary delays due to train interactions. This paper presents a stability analysis approach based on stochastic max-plus linear system theory. Stochastic counterparts of well-established concepts from the deterministic max-plus stability analysis are proposed, like timetable stability and realizability. General probability distributions can be used to model the primary stochastic behaviour of process times, while delay propagation due to timetable and infrastructure constraints are computed from the stochastic recursive system equations. Recently developed powerful algorithms can be utilized to analyse and improve large-scale stochastic systems, and to establish the amount of stochastic variations that a timetable can absorb without external control.